Trey Burton added 91 rushing yards on only three carries and Driskel ran for 81 yards on eight attempts.
Tennessee linebacker A.J. Johnson added a 1-yard touchdown run out of the Wildcat formation for the Volunteers, who were outgained 152-5 in the fourth quarter.
Tennessee (2-1, 0-1) seemed intent on ending the streak for much of the night in front of a sellout crowd of 102,455 at Neyland Stadium. The Vols led 20-13 and had a chance to take a double-digit advantage midway through the third quarter after an unsuccessful Florida fake punt attempt gave the Vols possession at the Gators' 47-yard line. Tennessee failed to capitalize and ended up punting into the end zone.
Florida dominated from that point on.
Burton, a fullback who often takes snaps out of the Wildcat formation, raced for an 80-yard touchdown on the first play of Florida's ensuing possession. That play was only Burton's second carry of the night. He went around left end for a 14-yard touchdown on his first attempt.
Florida got the ball back when Matt Elam picked off Tyler Bray's pass intended for Justin Hunter. Driskel then put the Gators in front for good with a 23-yard touchdown pass to Jordan Reed. The touchdown was set up by a 45-yard run from Gillislee, who had gained 27 yards on 11 carries up to that point.
The Vols never recovered. A Tennessee offense that had moved the ball well for the first 2 1/2 quarters could do nothing right the rest of the evening. And a Florida offense that had taken a grind-it-out approach for the first two weeks of the season started delivering big play after big play.
Jeff Driskel and Frankie Hammond connected on a 75-yard touchdown that extended Florida's lead to 34-20 with 9:55 left in the game. Caleb Sturgis' third field goal of the game - a 49-yarder - closed the scoring with 6:44 remaining.
Florida came from behind in the second half to win an SEC road game for the second consecutive week. The Gators had won 20-17 at Texas A&M last week after trailing 17-10 at halftime. Florida had lost all five times it had trailed at halftime last season.
This also marked the first time Tennessee had lost a game it led at halftime since Derek Dooley took over as the Vols' coach in 2010. Through the first 27 games of Dooley's tenure, the Vols had been 13-0 when leading at halftime and 0-14 when tied or behind at the midway point.